Complex viruses are assembled from simple protein subunits by sequential and irreversible assembly. During genome packaging in bacteriophages, a powerful molecular motor assembles at the special portal vertex of an empty prohead to initiate packaging. The capsid expands after about 10%-25% of the genome is packaged. When the head is full, the motor cuts the concatemeric DNA and dissociates from the head. Conformational changes, particularly in the portal, are thought to drive these sequential transitions. We found that the phage T4 packaging machine is highly promiscuous, translocating DNA into finished phage heads as well as into proheads. Optical tweezers experiments show that single motors can force exogenous DNA into phage heads at the same rate as into proheads. Single molecule fluorescence measurements demonstrate that phage heads undergo repeated initiations, packaging multiple DNA molecules into the same head. These results suggest that the phage DNA packaging machine has unusual conformational plasticity, powering DNA into an apparently passive capsid receptacle, including the highly stable virus shell, until it is full. These features probably led to the evolution of viral genomes that fit capsid volume, a strikingly common phenomenon in double-stranded DNA viruses, and will potentially allow design of a novel class of nanocapsid delivery vehicles.